The Great Pyramid: It's Divine Message - Chapter II

¶ 135. PLATE XIII. EQUAL AREAS.

A rectangle constructed of length equal to the circumference of any given circle and of breadth equal to the diameter of the given circle is in area equal to four times the area of the given circle.

The ancients formulated their metrological systems upon the above relations and the year circle of 3652.42 Primitive inches in circumference — a representation of the solar year to the scale of 10 P inches to the day. This gave a circle of 1162.6 diameter. Four of such circles are represented as A, B, C, and D on Plate XIII. EFGH is the corresponding rectangle of equal area. Its length FH = EG = 3652.42 P”., and its breadth EF = GH = 1162.6 P.”. KLMN is the corresponding square of equal area. Area KLMN = (3652.42 x 1162.6) square P. inches. Side of square = KL = LN = = 2060.66 P.”. The three equal areas — 4 circles, rectangle, and square — were defined as the aroura, the great unit of square measure.

Each of the circles A, B, C, and D falls precisely internal to the outer circle of stones at Stonehenge.

The circumference of 3652.42 P” was divided into 200 circumferential cubits of 18.2621 P”. The diameter of 1162.6 P” was divided into 100 diametric feet of 11.626 P”. The side of the square of equal area (the aroura) was divided into 100 common cubits of 20.6066 P”. This supplied three systems of linear units — a system of circumferential units, a system of diametric units, and a system for the linear measurement of straight line plane figures. Each system has its own cubits, feet, and digits.

The three systems were therefore derived from a system of Primitive Inches. The Primitive inch was of the value of 1.0011 British inches. Accidentally or intentionally, this is a 500 millionth part of the Earth’s Polar Diameter.

The derived systems were originally invented for common use for the purpose of avoiding calculations involving it. Simple formulae connected the three systems. The result of one such simple calculation is shown in the lower portion of Plate XIII. Here the strip of the aroura square is equal to the area of the segment of the circle of radius 50 diametric feet and of arc length 12 circumferential feet. Typical formulas for the calculations are given in ¶ I37C.

The original or basal system and the three derived systems are illustrated to comparative scale on Plate XV (refer ¶¶ 137, I37a, b and c for description, formulae, and worked examples).

¶ 136. PLATE XIV. GEOMETRICAL ANALOGY.

As a corollary of the relationship of ¶135, the area of a quadrant of a given circle = Length of Quadrant arc × ½ radius. This defines the area of a triangle of area equal to the quadrant area. The perpendicular height of the triangle = the quadrant radius and the base of the triangle = the length of the quadrant arc.

Plate XIV shows the relationship for the case of two similar isosceles triangles mOn and M₁O₁N₁., and two quadrants MON and M¹O¹N^1.

The base angle of both triangles is the base angle of the Great Pyramid’s right vertical section, 51°-51’-14”.3.

The conception underlying this representation in the Pyramid is that the Isosceles triangle of area equal to the quadrant area is constructed from the development of the quadrant arc on its mid-tangent. Thus on left hand side of middle Plate XIV, Q is the middlepoint of the quadrant arc MQN and mQn is the tangent at mid-point Q. The quadrant arc MQN is developed on to the tangent mQn, so that when QM is straightened out along Qm, M gives the point m, and when QN is straightened out along Qn, N gives the point n. The process is illustrated on the bottom left hand figure. Hence QM = Qm, QN= Qn, and MQN = mQn. Joining Om and On, we find that although the area has been distorted and the definitive linear dimensions retained,— OQ common to both, and MQN = mQn — the area of the isosceles triangle mOn is nevertheless equal to the area of the quadrant OMQNO. When one comes to think of it, this is a very remarkable and simple property. It is, however, a property that is seldom conceived in tangible form by the mathematician.

The same conception and simple relationship extend to sectors and triangles. Similar sectors, when developed, give similar triangles, each of area equal to the area of its sector, and of definitive dimensions equal to the definitive dimensions of its sector.

¶ 137. PLATE XV. COMPARISON OF ANCIENT SCALES OF MEASUREMENT.

The systems of measures briefly described in ¶ 135 are illustrated to comparative (reduced) scale on Plate XV. ¶ 135 described how the circumferential cubit, the diametric foot, and the common cubit were derived from the original linear unit, the Primitive Inch. Direct diagrammatic illustration is given in the lower portion of Plate XV. This gives a representation (right hand figure) of an Egyptian cubit rod noted by Professor Petrie.¹ Its length is the circumferential cubit, and on it is marked off the length of the diametric half-foot. The right hand figure illustrates the manner of direct derivation from the year circle geometry and the original Primitive inch.

THE DIAMETRIC SCALE:—

With the derived diametric foot (11.626 P”) as basis, this was divided off into 16 digits— each of value 0.7266 P”. The number of digits is from Petrie.¹

Whereas one and a half diametric feet contained 24 diametric digits, the diametric cubit “ was reckoned to contain 25 diametric digits. This is the real origin of “the well-known ratio of 25 : 24,” noted by Petrie.¹ (Refer ¶¶ 88-90.)

THE CIRCUMFERENTIAL SCALE:—

With the derived circumferential cubit (18.2621 P”) as basis, this was divided off, like the diametric cubit, into 25 digits — each of value 0.7305 P”. The number of digits is from Petrie.¹ The circumferential cubit also contained one and a half circumferential feet.¹ Petrie here remarks that this foot “although very well known in literature, is but rarely found ............... The Greek system, however, adopted this foot as a basis for decimal multiplication.” (Refer ¶¶ 88 and 89).

THE LINEAR SCALE FOR SIDES OF RECTILINEAR AREAS:—

With the derived cubit (20.6066 P”) as basis, this was divided off into 32 digits — each of value 0.644 P”. The number of digits in the early Babylonian and Egyptian examples of this cubit is from Petrie.2 Petrie explains that the later division into 28 digits was due to a confusion of this system with the systems herein defined as diametric and circumferential. Thus, 28 circumferential digits = 20.454 P”, closely approximating to the true value of 20.6066 P” for the common cubit.

¹‘Enc. Brit, (nth Edit.), Vol. xxviii, p. 483 c.

¶ 137a. THE ALGEBRAIC RELATIONSHIP OF UNITS. (PLATE XV.)

If any one value — A B b, d, H, h, L, or λ — be given, all the other values can be found directly from the formulae I to IV.

If any one of the values D, δ, or β is then required, it can be derived from formulae (1) and (2).

If any one of the values D, δ, or β is given, its value in terms of d — for D and δ — and in terms of b or B for β, can be found from formulae (i) and (2), and thereafter substituted in formulae I to IV, as

¹Brit, (nth Edit.), Vol. xxviii, p. 483 b.
²Ibid. p. 482 d.

¶ 137b. EXAMPLES OF SIMPLE RELATIONS. (PLATE XV).

One important relation is obtained from the formulas as follows :—
A given diameter = δ diametric digits.

From Formula (II):—
Length of side of square of equal area, in digits of common cubit = λ = 16d.
From (I) : — δ = 16d.
Hence λ = 8.

Otherwise expressed, the length of side of the square of area equal to the area of a given circle contains the same number of digits of the common cubit as the diameter of the given circle contains diametric digits.

A worked example of the above is given for a circle of diameter measuring 2,000 diametric digits.

¶ 137c. THE SIMPLE CALCULATIONS FOR AREAS of SECTORS AND SEGMENTS OF CIRCLES.

Let m = No. of Circumferential Cubits in a given Sector arc, of diameter d diametric feet, for circle of B
circumferential cubits.

Otherwise expressed, the area of a given sector in common square cubits is equal to one-eighth the product of the number of circumferential cubits in the sector arc and the number of diametric feet in the diameter of the circle ; or, is equal to a quarter of the product of the number of circumferential cubits in the sector arc and the number of diametric feet in the radius of the circle.

To obtain the area of the segment in the given sector, in common square cubits, deduct the area of the isosceles triangle of the given sector from the area of the sector as above obtained in common square cubits.

¶ 138. PLATE XVI. CHART SHOWING THE GEOMETRICAL, ASTRONOMICAL, AND NUMERICAL BASES OF THE FICTITIOUS CHRONOLOGIES OF THE ANCIENT EGYPTIAN KING LISTS.

General remarks:—
The chart is a record of facts that have been long in existence—in some cases for several thousand years. The elements that are distinctly new are the co-ordination of these facts and the self-evident origin and significance of the facts revealed by this co-ordination.

The outstanding new facts derived from the statement of the chart are the following :—

(1) That the Egyptian King Lists of the Egyptian Priest, Manetho, do not contain a true statement of
ancient Egyptian Chronology, ¶¶ 92, 118 and 119.)
(2) That prior to the 3rd century B.C., the Egyptians knew nothing concerning the hypothesis now
adopted as the basis of modern Egyptological chronology, (¶ 98 and Appendix.)
(3) That the King Lists contain a written record of the numerical values of all the external linear and
angular measurements of a Standard Pyramid ¶¶ 93, 95-99, 118 and 119), in terms of units
specified in the Lists as of values equal to 1.0011 British inches and 20.63 British inches
respectively, ¶ 94.)
(4) That the Standard Pyramid of the Egyptian King Lists is the Great Pyramid of Gizeh. (¶¶ 94,
99- 101 and 118.)

The complete statement of Manetho’s Divine Dynasties is as given in Table A of chart. This is precisely as stated by Sir Ernest Budge, “ Book of Kings,” Vol. I, pp. lx and lxi.

The detailed statement of Manetho’s Human Dynasties is as given in the Appendix. This is precisely as stated in Baron Bunsen’s Greek and Latin Text (“ Egypt’s Place,” Vol. I, Appendix), for the versions of Africanus and Eusebius, and in Cory’s “ Fragments “ (Hodge’s Edition, 1876). The other lists are preserved in the same works. Statements of Manetho’s Lists also appear in Budge’s “ Book of Kings,” Vol.I, his “ History of Egypt,” Vol. I, in Sayce’s “ Ancient Empires of the East “ (Appendix), and in the various volumes of Petrie’s

“ History of Egypt.” These, however, generally omit some important details and statements peculiar to the Version of Africanus. Budge’s statement (“Book of Kings,” Vol. I) of the basal totals of years for the Version of Eusebius for Manetho’s Book I, II and III has been adopted in the chart (Table B). The stated totals for the same books, according to the Version of Africanus, have been adopted from Cory in the chart (Table B).

¶ 138a. SOME DETAILS CONCERNING THE VERSION OF AFRICANUS.

Four features affecting the statement of the Version of Africanus in Tables B and C call for special remark.

^lAncient Egypt, 1914, p. 32. 1916, p 150.

The summation of Plate XVI, Table A indicates that the statement of Africanus concerning the 990 years is to be similarly explained. 990 years added to 24,837 years, the duration of the Divine Dynasties, give 25,827 years, the sum of the Pyramid’s base diagonals. 990 years added to the 4,611 years of Eusebius for the human kings, give the 5,601 years of Africanus for the human kings.

¶ 139. PLATE XVII. DIAGRAMMATIC REPRESENTATION OF PROFESSOR PETRIE’S-RECONSTRUCTION, AND OF THE NEW RECONSTRUCTION, OF THE GREAT PYRAMID FROM PETRIE’S SURVEY.

This Plate fully explains itself. One item, however, may require amplification; the relation between the dimensions on diagrams of Plate XVII, stated in Primitive or Pyramid inches, and the dimensions according to Petrie’s survey in British inches.

Petrie’s survey for the mean square side defining the corners of the existing core surface base gave a length of 9001.5 B” ± 1.0 B”. Reduced to Pyramid inches (on basis of ¶¶ 81, 94 and 101), this is 8991.6 P”, ± 1.0 P” or, as stated in round numbers of inches as on Plate XVII, Figs. A, B and C, 8,991 P”.

Petrie’s mean distance between the centres of two opposite sides of the core masonry : base—i.e., as along line of AB, Figs. A, B and C; or on Section AB, Figs, a, b and c—gave 8,929 B”, or in round numbers of P. inches, as on all figs, of Plate XVII, 8,919 P”.

The other relations defining the casing base square, and its central hollowing in, are as given in ¶ 99-101.

As to what Petrie means by a core plane face, the reader is referred to Plate XVIII and ¶ 140.

It is unfortunate that Professor Petrie, in observing the core masonry hollowing, did not extend the same feature to the restoration of the casing. By reason of this unfortunate , omission, scientists for 42 years have been led to believe that the theory of the late Astronomer Royal for Scotland—Professor Piazzi Smyth—requiring a Great Pyramid base circuit of 36,524 Pyramid inches, was nothing more than a delusion. It is equally unfortunate that Professor Smyth saddled his theory with corollaries and side issues rightly deemed by his scientific contemporaries to be fallacious.

¶ 140. PLATE XVIII. DIAGRAMMATIC PERSPECTIVE VIEW, ILLUSTRATING FEATURES OF GREAT PYRAMID’S CORE MASONRY.

As explained on the Plate, the hollowing-in of the core face escarpments, and the depths of courses are considerably exaggerated. In consequence of the latter, the number of courses is reduced. The thicker 5^th course, however, gives a general idea of the appearance of the 35^th course. The view illustrates the Pyramid’s appearance prior to the addition of the casing.

What Petrie means by a core plane face is defined by the plane geometrical surfaces cCBb and bBAa. Petrie’s core plane base is the actual square defined by the corner points , C, B, A and D (the last unseen). This base square was obtained by sighting down from c, b, a and d, along the line of the stepped (arris) edges, cC, bB, aA and dD (the last unseen). , The core base square is defined by the straight lines CB, BA, AD (unseen) and DC (unseen).

The hollowed-in effect is defined on the base by the lines CHGB and BFEA. HG , and FE are each about 36 inches horizontally internal to the square base sides CB and BA. HG and FE were obtained by sighting down the stepped core courses from c to H and b , to G; and from b to F and a to E.

Petrie states, “The form of the present rough core masonry of the Pyramid is capable of being very closely estimated. By looking across a face of the Pyramid, either up an , edge, across the middle of the face, or even along near the base, the mean optical plane, J which could touch the most prominent points of all the stones, may be found with an average variation at different times of only 1.0 inch. I therefore carefully fixed, by nine observations at each corner of each face, where the mean plane of each face would fall on the socket floors; using a straight rod as a guide to the eye in estimating. On reducing these observations to give the mean form of the core planes at the pavement level, it came out thus :—

On pp. 43-44, Petrie then states as to “the faces of the core masonry being very distinctly hollowed.” “This hollowing,” he continues, “is a striking feature ; and beside the general curve of the face, each side has a sort of groove specially down the middle of the face .............. The whole of the hollowing was estimated at 37 B” on the N face................... “

¹Pyds. and Temples of Gizeh, pp. 37, 38.